11 THE SPACES L1 AND L2 171 for some integer k between 0 and n — 1 ; show that fdp ^ 0 by using (c) above.) J Ejj(,f) Hence deduce the maximal ergodic theorem of Section 13.9, Problem 12. 17. Suppose that U satisfies the conditions of Problem 16 and also that for each / > 0 the relation I/O) | <£ t almost everywhere in X implies that | (U •/)(*) | rg / almost every- where. (This will be the case whenever ^ is bounded and U - c = c for each constant c.) (a) Show that, for each function /e &£ and each t > 0, (1) (U-f-t)+<U*(f~ty almost everywhere. (Consider the function which is equal to /when \f(x)\ £*/, to t when f(x) > r, and to —t when f(x) < --/.) (b) Show that, for each function fe&^n&g, the function U •/ belongs to ^ for 1 <: p <s + oo and that (Deduce from (1 3.21 .9) that for each/e &£ n JSfff , the function (*, /) >-> tp~2(f(x) - 1)+ is integrable relative to the measure on X x ]0, + oo [ which is the product of /x and Lebesgue measure, and then integrate the inequality tp'2(u •/- o+ < ^-2(/- /)+.) Deduce that (/can be extended to an endomorphism of norm ^ 1 of each of the spaces J&& . Denote this extension also by U. (c) Let /e J$?& for some p € [1, + oo[. For each t > 0 and each integer n J> 0, let E,,,t(/) denote the set of points x e X such that /(*) + «/ ' /)(*) ~l~ - ' • 4- (Un- 1 • f)(x) > nt, and let Et(/) denote the union of the EM,r(/). Show that each of the sets En,t(f) is integrable and that (Observe that EM(/) is contained in the set of points x e X satisfying one or other of the n relations (Uk • f)(x) > t, where 0 < k ^ n — 1 ; then continue as in Problem 16, by remarking that (U-f+U2-f+--- + Uk-l'f-(k + W g U- (/H- (/•/+••• + Uk •f-(k+l)t)+.) (d) If /e J2?i , deduce from (c) that for all / > 0. 18. Suppose that U satisfies the conditions of Problem 16. Let f,g e J§f£ be such that (a) Let £ be a real number > 0, and let An be the set of x e X which satisfy the inequality (Un •/)(*) > e(ff(x) + (U- g)(x) -f • • • -f (Un^r/™ 1)— /)->0 as t-*Q. (Let /== u + w where u and v are real; observe